Local times and almost sure convergence of semi-martingales

A NNALES DE L ’I. H. P., SECTION B

B. R AJEEV

M. Y ^OR

Local times and almost sure convergence of semi-martingales

Annales de l’I. H. P., section B, tome 31, n

^o

4 (1995), p. 653-667

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653

Local times and almost

sure

convergence of semi-martingales

B. RAJEEV

Stat-Math Division, Indian Statistical Institute 203 B.T. Road, Calcutta, 700 035, ^India

M. YOR

University P.-et-M.-Curie, Laboratoire de Probabilités Tour 56, ^3e etage, 4, place Jussieu, 75252 Paris Cedex 05.

Vol. 31, n° 4, 1995, p. 667 Probabilités et Statistiques

INTRODUCTION

In this paper, ^we

study

^the connection between the almost sure

convergence of

semi-martingales

^{and the}

asymptotic

behaviour of their local times. Our

study

^{is motivated}

by

^the

following example.

Let h

(t)

^{be a}

non-negative decreasing

function and

(Bt)

^{a Brownian}

motion. Let

Xt = h (t) ^Bt.

If

Xt

converges ^{as t - oo,} then it must converge ^to zero, because

{t: ^{Bt =} 0~

is unbounded. We expect ^{that this}

property gets

^reflected

in the behavior of the local time of X at zero, which is

easily

^{seen to}

be

equal

^to the process

( s ) dls),

^,

^{where l t}

is the local time of B

at zero. In section

1,

^{we state} necessary and sufficient conditions so that lim h

(t) ^{Bt =}

^{0 and lim}

h ^{(s) dls}

^{oo, in the case when h is a}

t--~oo

/

deterministic function. These results are due to Jeulin and Yor

(see [6], Proposition ¹⁵⁾

^{and C.} Donati-Martin

(see [3],

p.

150) respectively.

Annales de l ’lnstitut Henri Poincaré - Probabilités et Statistiques - ^0246-0203

654 B. RAJEEV AND M. YOR

Now, ^the

integration by

parts ^formula

gives

^us:

and the effect of the drift term is to

pull

the process towards the

origin.

We make this notion

precise

^{below in Section}

^2,

^{where we}

study

^subsets

of the

sample space 03A9

^{on which the drift term of a}

given semi-martingale

has this

property.

^{In the case of a} continuous

martingale,

the almost sure

convergence of the

martingale,

^its local time and its

quadratic

^variation

processes ^are all

equivalent.

^{In the case of the}

semi-martingales

^{which we}

study,

the situation is similar and

yet

^{there are some} subtle differences.

1. BROWNIAN ASYMPTOTICS

Consider a continuous local

martingale (Mt)t&#x3E;o, ^Mo

^{= 0 a.s. Let}

denote

a jointly

continuous version of its local times and

((M)t)

^its

quadratic

^variation process. The

following

^lemma is well known

(see,

e.g.

[12], Prop. (1.8),

p.

171)

and follows

by writing

^{M as a time}

change

of Brownian motion.

LEMMA 1.1. - For _{every x E}

R,

^the

following

^{sets are almost}

surely equal:

(i) {03C9 :

^lim

^Mt exists}

(ii) {cc; :

^lim

^Mt

^{exists and is}

finite}

(iii) {03C9 : M~~ 00}

(iv) {w: L~ oo}.

Now, ^{consider a} Brownian motion

( Bt ) t &#x3E; o ^starting

^{from zero. Let}

gt ⁼

B~

⁼

0} and {Hu, u

&#x3E;

0}

^a

^locally

^bounded

^previsible

process; ^let

MH

⁼

Hgt ^Bt.

^It follows from the

"Balayage

formula"

(Azema-

Yor

[1],

^Yor

[8])

^{that is a} continuous local

martingale

^{and it} is easy to see that its local time at 0 is

t&#x3E;o

where is the local time of

(Bt)

^{at the}

^origin.

^{We have the}

^following

consequence of Lemma 1.1.

PROPOSITION 1.2. - The

following

^are

equivalent:

(i)

^lim

(Hgt Bt)

^{= 0 a.s.,}

(ii)

^{oo a.s.,}

Annales de l ’lnstitut Henri Poincaré - Probabilités et Statistiques

We now

specialise

^{to the case when}

(s),

where h is a non-

negative decreasing

deterministic function. It is easy ^{to see} that the process

is

^{also the} local time at the

origin

^{of the}

semi-martingale

wo

A&#x3E;o

Xt

^{= h}

(t) ^Bt.

^It is well known that the measure

(induced by

^the

increasing

process

(It)to ^is,

^almost

surely, singular

^{w.r.t. the}

Lebesgue

^{measure on}

[0, oo).

^The

following

^result

(first proved by

Donati-Martin

[3],

^{when h} is

smooth)

is nonetheless true.

THEOREM 1.3. - The

following

^are

equivalent.

The

proof

^{of this theorem}

depends

^{on a result of Jeulin}

[4]

^also Jeulin

[5] and,

^{for some}

applications,

^Xue

^[7]).

^{We state and} prove it for

completeness.

LEMMA 1.4. - Let (Rt)t~0 be

a positive measurable process such

^that

1)

The law 03BD

of Rt

^does not depend on t.

2)~({0})

^{= 0}

3)

^E

(R,)

^oo.

..00

Then, for

any

positive Radon measure

^on

R, ~0 ^{d (t)}

^oo

^iff

~0 ^{Rtd (t)}

^{00 a.s.}

The

part

is clear: if

~0 ^{d (t)}

^{oo, then in fact}

^by

condition

3),

^E

~0 ^{Rtd (t)}

^oo.

Conversely, let ~00 ^{Rtd (t)}

^{oo a.s. and let n} be such that P

(Jn)

&#x3E; 0

7o

where

Jn = {03C9 : ~0 Rt ^{d (t)} ~ n}. ^Then,

656 B. RAJEEV AND M. YOR

Since &#x3E; 0 and v

[0, uJ

^{- 0 as u - 0}

by

^Condition

2),

^{the last}

integral

is in fact

strictly positive,

say

equal

^{to an. It} follows that:

and the

proof

^is

complete.

Proof of

^Theorem 1.3. - Since E ⁼ c

0

^{where c is a constant}

not

depending

^{on s, it is} easy ^{to see}

(since

^{h is}

deterministic)

^that

E

="2 ~. ^{Clearly, (ii)~(i).}

To go ^the other way, suppose that

(i)

^{holds. We will} show that the ^two functions h

(t) ~

^and

fo ^q7

^dh

^(s)

^{converge to a} finite limit as t ^- oo.

It follows from the

equation

that

We now show:

Since both terms in the LHS are non

negative

^and since the RHS converges

by assumption,

it follows that

100 ^{ls (-dh9)}

^{oo a.s.}

^Now,

Lemma 1.4

applied

^to

^Rt = ~

^and

(8) = -.8

^dh

(s)

^{shows that}

We then show: lim h

(t) fi

^oo. Since h is

decreasing, h (t) |Bt| ^~

h

(gt)|Bt|, where gt

⁼

Bs

⁼

0}.

^From

Proposition ^1.2,

^it

follows that h

(t)

^{0 a.s. Since}

(ht Bt)

^are

gaussian

^random

variables,

this

implies

^that

ht Bt -

^{0 in i.e. h}

(t)

^{0 and the}

proof

^is

complete.

Remark 1.5. - Let H be a bounded

previsible

process. From

Proposition ^1.2,

^{is a}

martingale

^{which is not}

uniformly

^inte-

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

grable,

unless it is

identically

^zero.

Consequently,

^if

/ /~ ^{) Hsl dls}

^{oo a.s.}

then

by Proposition 1.2 ~

^{oo a.s. but E}

Jo ^ds

^=00.

Remark 1.6. - There are situations where the conclusions of Lemma 1.4

are true, but

ERt

^{= oo.}

Let gs

⁼ sup

{u ~ s : ^Bu

⁼

O}.

^{Consider the}

pair (ds))

^where

^Rs

⁼ s203B1, ^a &#x3E;

1/2 and (ds)

⁼ -2

(s).

~s ^{° ~}

Then ~ ~ ^(ds)

^{oo and from}

Proposition

^1.2

applied

^to

^Hs

⁼

o

cx&#x3E;

~

follows

that ~ ~00 ^(ds)

^{00. But this} conclusion cannot be obtained from Lemma 1.4 because

ER1

⁼ oo, since ₉₁ is arcsine distributed.

In the next

Theorem,

^we

reproduce

^a result of Jeulin and Yor

[6]

^which

gives

^a necessary and sufficient condition for h

(t) ^Bt

^{to converge to zero}

almost

surely

^{when h is a} deterministic function.

Here,

^we consider the

particular

^case where h is

decreasing.

THEOREM 1.7. -

(t)

^{be a non}

negative decreasing

deterministic

function.

Then, ^the

following

^are

equivalent:

(i)

^{lim h}

(t) ^Bt

^{= 0 a.s.;}

(ii) for

every ~ &#x3E; 0, ~1 exp(-~ th2(t))

dt t

^~.

Proof. -

^Let

B~

⁼

sup ) ^Then,

^condition

⁽ⁱⁱ⁾

^is

equivalent

^to

~~ / &#x3E; ~2"~B j oo

^for every ~ &#x3E; ^0.

This follows from the

inequalities:

^{for e} &#x3E; 0,

and

We now show that

(ii’)

^{holds iff}

(i)

^holds.

31, n°

658 B. RAJEEV AND M. YOR

Let

Vn

⁼

2-n/2

_sup

I ^{Bt -}

^{It is} easy ^{to see,}

by

^Borel-

Cantelli arguments, ^that

(ii’)

^is

equivalent

^{to lim}

(2n)

^{= 0 and}

that

(ii’) implies lim (2~)

^{= 0. In}

particular,

^{we remark that:}

Now, (ii~)=~(i)

follows from the above observations and from the

inequalities

Conversely,

^if

(i) holds,

^{then lim}

Vi h (t)

^{= 0. Moreover,}

(Vn )

are

independent

^{and have the same law as}

B;

⁼

sup

^{. Hence}

lim

2n/2 Vn h (2’~)

^{= 0 and}

^(ii’)

^holds.

Section 2. - In this

section,

^we prove ^a version of Lemma 1.1 for semi-

martingales.

^{This is Theorem 2.4. Our}

analysis

^will be restricted to the class of

semi-martingales (Xt)

^with

L ^I

^{oo, a.s. for all}

t &#x3E; 0.

^These

st

can be written in the form

Xt

⁼

Xo + Mt

^where

(Mt)

^{is a continuous}

local

martingale

^{and a} process of finite variation. For a

semi-martingale X,

^this property will be assumed to hold for the rest of the _paper.

We recall the Tanaka formula for the

semi-martingale

^{X : for a E}

R,

where

and

(Lt )t&#x3E;o

^{is the local time at a of}

^X,

^{which is an}

increasing

^continuous

process,

supported

^{on the set}

{s : ^Xs-

⁼

^Xs

⁼

a};

^{for a} discussion of these

formulae,

^see _e.g. ^Yor

([ 11 ],

p.

20).

^{For the}

semi-martingales

^which

we

consider,

^{there is a} version of the process

0,

^{a E}

R)

^which

is

jointly right

continuous in

(t, a)

and has left limits

(see

^Yor

[9]).

^This

property

^is

important

ⁱⁿ what follows and we will

always

^take

equations ( 1 )

^and

(2)

^{to be true outside a} null set, for all t

&#x3E;

^{0 and for all a E R.}

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

Let

V+ (t)

⁼ &#x3E;O}

dVs

^{and V-}

(t)

⁼

/ 7{x _

^{~0}dVs. The finite}

variation

part

^{V of the} processes considered in

Examples

^{1 and 2 have the}

property

^that

V+

^is

decreasing

^{and V- is}

increasing.

^{For a}

semi-martingale

X ⁼

Xo

+ M +

V,

^we define the

following

subsets of S2:

{cc; : V+

^is

decreasing

^{and V- is}

increasing}

v± _ {w: V+

^is

increasing

^{and V- is}

decreasing}

v+ _ {w: V+,

^{V- are}

increasing}

{w: V+,

^{V- are}

decreasing}.

Of course, these sets do not

necessarily partition

^{H. We shall use the}

notation

%00 (resp. Xoo)

^{for lim inf}

Xt (resp. lim sup Xt).

^{We shall denote}

t-+oo

lim

Xt by Xoo

whenever it exists. In this

notation, {w : ^{Xoo E} (a, b)}

means

{03C9:

^lim

^Xt

^{exists and}

^belongs

^to

(a, &#x26;)}.

^The

^following

^lemma

will be used in the

proof

^{of our main result}

(Theorem 2.4).

LEMMA 2.1. - Let -~ a b 00. Then:

(i)

^On

the set w : ^{(XS- ) dYs}

^is

decreasing},

^we

^{have, for}

all c E

(a, b],

^almost

surely,

and

t

(ii)

^{On the set}

t0I(a,

^b]

^{(Xs- ) dVs}

^is

increasing},

^{we have}

^for

^all

c E

~a, b),

^almost

surely,

and

Vol. 31, n° ^4-1995.

660 B. RAJEEV AND M. YOR

Proof - (i) Suppose

^that

/ I(a, b] (Xs-) ^dVs

^is

decreasing. Then,

^for

c e

(a, &#x26;]

^we

^have,

^from

equation ^(1),

^that

Note that the LHS of

equation ⁽³⁾

is bounded

by 2 (b - a).

^{Now if}

L~

^oo,

it follows that the

decreasing function 0 I(a,c] (Xs- ) dVs - 1 2 ^Lct

^{has a}

finite

limit,

^{as also}

t0 ^{7(a,c] (Xs-) dMs (this}

follows from Lemma

1.1).

Thus

Lc~

^oo,

proving

^the first inclusion in

(i).

^{Now the LHS of}

equation (3)

^{has a finite limit as t - oo. In}

particular,

^{it has a} finite limit when

c ⁼ b. But this means

precisely

^that

~oo ~

^{b or}

X~

^{a or}

(a, b).

(ii)

^The

proof

is similar to that of

(i) using

^the

equation

for all

COROLLARY 2.2.

Proof. -

^It follows from the results in

Yoeurp ^[10]

^that

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

Hence if

[X, X]~ (w)

^{oo, then}

L~ (w)

^{oo for}

^{all x}

^N

(w)

^C

^R,

A

(N (w) )

^{= 0,} where A is the

Lebesgue

^{measure. It follows from Fubini’s}

theorem that there exists N c

R, A (N)

^{= 0, such that:}

Choose _~,

~ 1~

^{6. If}

~0 / ^{(X~-) ~}

is monotone, then so

~(a~,6~] (~s-) ^~

^{for every ~.} It follows from Lemma 2.1 that if

[X, X]~ ((03C9)

^{~, then or}

^Xoo

^or

^X~

^e

(an, bn)

for all ?~. This means that or

Xoo

^{a or}

(o~? ~).

^Li

COROLLARY 2.3. - On the set

03BD-+

and

In

particular,

^{on the set}

v+

for

every ^a.

THEOREM 2.4. - Let

(Xt)

^{be a}

semi-martingale

^with

^Xt

⁼

Xo

⁺

Mt

+

V’;;

where

(Mt)

^{is a} continuous local

martingale

^and

(Vt)

^{a process}

^{of finite}

variation. Then, ^{we have the}

following:

a) (i)

^{On the set}

v+, ^for

^all

x ~

^0,

and

Vol. 31, n° 4-1995.

662 B. RAJEEV AND M. YOR

holds on each

of

^{the sets}

v± , v+ ,

^{v- .}

(ii) {03C9 :

^lim

^Xt

⁼

X~ exists, C {03C9 : [X, X]~

~} C _{{03C9 :}

^lim

Xt exists}

^{holds on each}

of

^{the sets}

v±, v+,

^v-.

Proof - a) (i) Suppose

^{0 a}

^b,

^{a, b E}

Q. Then, by

^{Lemma 2.1}

(i),

Letting

^a

1

0,

b i

^{oo, we} get

Similarly,

from Lemma 2.1

(ii),

it follows that

It follows that on the set

v-+

Conversely,

suppose w

~ {lim ^Xt exists}

ⁿ

v+ . Since {lim ^Xt exists} =

^E

(0, oo)}

^{U E}

(-0o, 0)}

^U

^0,

it is sufficient to show that on each of the sets in the

RHS, L~ (w)

^o

for w and

a #

^0.

If w E

(0, oo)}

ⁿ

~,

^{then from the}

decomposition ^Xt

⁼

Xo

⁺

Mt

+

V+ (t)

^{+ V-}

(t),

it follows that

V-, V+

^{and hence M converge}

to a finite limit and hence from

equation ( 1 )

^that

L ~

^oo,

Annales de l ’lnstitut Henri Poincaré - Probabilités et Statistiques

Similarly,

^from

equation (2),

^it follows that

If w ⁼

0,

ⁿ

v+ ,

^then

since a ~

^{0, the} process does ^not visit

a after a finite time and there are no

jumps

^{of X which cross a.}

^Then,

since

Lf

⁼

Lf

⁺

If,

^it follows that

Lf

^{does not} increase after a finite time i. e.

La~

^o.

Thus {X~

⁼

0, ~ 03BD-+ ~ {03C9 : ^La~

^{and the}

proof

^of

a) (i)

^is

complete.

a) (ii)

To prove the first inclusion in

a) (ii),

^{note that as in the}

proof

of

a) (i),

If w E RHS

above,

^{then of course oo} and

Thus, {(~ :

^I

Xo # 0, ~’ _{{~ :}

^I

_[X, X]~

To prove the second inclusion in

a) (ii),

^{let first} 0 ^a b. From

Corollary

^{2.2 we} get,

Now

letting a ~,

0,

b i

^{oo, we} get

Similarly, by applying Corollary

^{2.2 to the case a b} 0, ^{we can} prove,

The second inclusion in

a) (ii)

follows from

(5)

^and

(6)

^and

completes

the

proof

^of

a).

b) (i).

^The

proof

^that

n {~ L~ {w:

^lim

^Xt exists}

^on

6~Q

the sets

~i, v t,

^~1 is similar to the

proof

^{for the set}

~,

^{described in}

^a) (i) using

^{Lemma 2.1. We will} prove the ^reverse inclusion on the set

v±,

the other two cases

being

^{similar. Fix b E R.}

Vol. 31, n° 4-1995.

664 B. RAJEEV AND M. YOR

Since

{03C9 :

^lim

^Xt exists} = {03C9 : ^Xoo

⁼

^0,

^U

^0, ±oo},

it suffices to show that each of the sets in the RHS is contained in

{w : Lb~ ~}

^on

^03BD+-.

^{For the}

set {w : ^0,

^the

proof

is similar to the

proof given

ⁱⁿ

^{a) (i).}

^{We will} show the inclusion

{w : ^Xoo

⁼

^0,

ⁿ

{w : L~ 00}

for every b ^E R.

Now, ^{if or if} 0, ^and

b ~ ^0,

^then

L~

^{does not}

increase after a finite time. In

particular, L ~

^{o. If}

^Xoo

^{= 0 and b = 0,}

then since on

~1, _V+ (t)

⁺

L~

^is

increasing,

^it

^follows, by letting t

^{~ o0}

in

equation ⁽³⁾

^that

L~

^oo.

b) (ii)

^{It is} easy ^{to see,}

using

^the

decomposition ^Xt

⁼

^{Xo + Mt +} V+ (t) +

V-

(t),

^{that if}

^0,

^then

^{M, V+,}

V- converge ^{to a} finite limit on

the set

v± .

^If

Xoo

⁼ 0, ^{then we} argue ^as in

b) (i)

and conclude that

M, V+,

^{V- converge to a} finite limit on the set

x/i. Thus,

^on

v±,

Now

exactly

^{as in case}

a) (ii),

^{we can show that the RHS set is contained}

in

{03C9 : [X, X]~ oo}.

The inclusion

{w: [X, X]~ ~}

ⁿ

{w:

^lim

^Xt exists}

^{is also}

^exactly

^{as in case}

^{a) (ii).}

^This

^completes

^the

proof

^of

^b)

^{and the}

proof

^{of the} theorem. D

Remark 2.5. - The inclusions in Theorem 2.4 viz:

can be strict. Consider for

example

the process

Xt

^{= h}

(t) ^Bt

^where

(Bt)

is a Brownian motion and h

(t)

^non

negative decreasing (t)

’" y’tlOgt

as t ^- oo .

Then, by

the law of the iterated

logarithm, Xt ~

^{0 a.s., but}

[X, X~~ _ ^{(X~~~, _ (s)}

^{ds = oo.} On the other

hand,

^if

/~) - -

as t ^- oo, then

{w: [X, X]~ 00}

⁼

^~, {~ : X~ ~ 0~ _

^0.

We now consider the local time

L~

^{on the set The}

^following

^lemma

shows that in this

situation, L~ plays

^a

special

^role.

LEMMA 2.6. - On the set

1I+,

~ _{{03C9 :}

_t--~~ ^lim

.Xt

exists and is

finite}

ⁿ

{03C9 : [X, X]~

^"

Annales de l’Institut Henri Poincaré Probabilités et Statistiques

Proof. -

From Lemma

2.1,

^it follows that for all b _E

R+,

^{on the set}

v+ , ~ w : Loa Loa oo}.

From Theorem 2.4

a) (i),

^it

follows that on

x/~,

Now, equations (1)

^and

(2) imply

that in fact on

~,

But the RHS set is the union of the sets

{w : ^X

⁼

0~

^and

~w : 0~ ±00}.

^{The latter set is}

already

^{contained in}

{w : [X, XJ 00 oo}, by

Theorem 2.4

a) (ii).

^{If w} E

0},

^we argue ^as

follows: _firstly,

it follows from

equations (3)

^and

(4)

^that

/ 7(o,

^a)

^{(for a}

&#x3E;

0)

and d

(for

^a

0)

^are both finite.

Since

Xoo = 0,

^it

implies

^{in fact that}

(M)oo

^{oo and} hence that the

martingale

part

Mt

converges.

Now,

from Tanaka’s

formula,

^{it follows} that

V+

^and V- converge ^{to a} finite limit and hence that

L ^aXs

^oo.

[X, X]oo

^{oo now follows.}

Remark 2.7. - We list below the

mutually

^exclusive

possibilities

^that

can occur on the set

1/

^with respect ^to the variables

L~, [X, X]~

and

Xoo.

^{Now with} the

help

of Theorems 1.3 and

1.7,

^{we can}

explicitly

compute functions h

(t)

^so that these

possibilities

^{are in} fact realised

by

^the

semi-martingale Xt

^{= h}

(t) ^Bt

^where

Bt

^{is a} Brownian motion.

a) L~

^oo,

[X, X]~

^{o and}

0, ±00.

^{This case does not arise}

for the

semi-martingale Xt = h (t) ^Bt.

Vol.

666 B. RAJEEV AND M. YOR

ACKNOWLEDGMENTS

1 )

^{This work was done} while the first author was

visiting

the Laboratoire de

Probabilites,

Universite de Paris

VI,

^{under an}

exchange

programme between CNRS

(France)

^{and NBHM}

(India).

^{He would like to} thank Prof.

P. A.

Meyer

^for

making

^{the visit}

possible.

2)

^Partial

_support

^{of the} research of the second author

by

^{NSF Grant No.}

DMS 8801808 is

gratefully acknowledged.

3)

^{We thank the} referee for his

suggestion that,

^for

clarity, only

^one

example

should be discussed in the Introduction.

Nonetheless,

^{we would like to} mention that the

inhomogeneous

^Markov

process

(Yt)

studied in Chan and Williams

[2]

and which satisfies the stochastic differential

equation dYt

^{= -F}

(x)

^{dt + ~}

(t) dBt

^{was one of}

the

original

motivations of our

study.

REFERENCES

[1] J. ^{AZÉMA and M.} YOR, En guise d’introduction. Astérisque 52-53. Soc. Math. France, 1978.

[2] ^{C. T.} CHAN and D. WILLIAMS, ^{An excursion} approach ^{to an} annealing problem, ^Math.

Proc. Camb. Phil. Soc., 105, ^1989.

[3] ^C. DONATI-MARTIN, Transformation de Fourier et Temps d’Occupation Browniens. Proba.

Th. and Rel. Fields, 88, 1991, pp. 137-166.

[4] ^T. JEULIN, Semi-martingales ^et grossissement ^de filtrations, Lecture Notes in Maths, 833, Springer-Verlag, ^1980.

[5] ^T. JEULIN, Sur la convergence absolue de certaines intégrales. ^Sém Proba XVI. Lect. Notes in Maths, Springer-Verlag, 920, 1982, _p. 248-255.

[6] ^{T. JEULIN and M.} YOR, Filtration des ponts ^{browniens et} équations différentielles

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1990.

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[8] ^{M. YOR, Sur le} balayage ^des semi-martingales continues, ^{Sém. de Prob.} XIII, ^Lecture Notes in Maths, Springer-Verlag, 721, ^1979.

[9] ^M. YOR, Sur la continuité des temps locaux associés à certaines semi-martingales.

Astérisque, 52-53, Soc. Math. France, ^1978.

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

[10] ^Ch. YOEURP, Compléments ^{sur les} temps ^{locaux et les} quasi-martingales, Astérisque 52-53,

Soc. Math. France, 1978.

[11] M. YOR, Rappels ^et préliminaires généraux. Astérisque 52-53, Soc. Math. France, 1978.

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Vol. 31, n° ^4-1995.